# Logic/math phrasing **URL:** \ **Category:** Factual Questions\ **Created:** [July 17, 2018, 3:32am UTC](https://boards.straightdope.com/t/logic-math-phrasing/818061 "2018-07-17T03:32:34Z")\ **Posts on this page:** 14\ **Page:** 1
**Author:** ![OldGuy](https://avatars.discourse-cdn.com/v4/letter/o/3bc359/32.png) [@OldGuy](https://boards.straightdope.com/u/OldGuy)\ **Post date:** [July 17, 2018, 3:32am UTC](https://boards.straightdope.com/t/logic-math-phrasing/818061/1 "2018-07-17T03:32:34Z")
Given three statements A B and C, is there any difference at all between an author’s claims: A and B are necessary and sufficient conditions for C. and A and B are individually necessary for C and jointly sufficient. The only thing I can think is if A and C are equivalent statements and B is necessary for both, the first claim is true, but maybe the second is not because B is not jointly needed given A, but that seems to be stretching the point. ---
**Author:** ![borschevsky](https://avatars.discourse-cdn.com/v4/letter/b/97f17d/32.png) [@borschevsky](https://boards.straightdope.com/u/borschevsky)\ **Post date:** [July 17, 2018, 4:11am UTC](https://boards.straightdope.com/t/logic-math-phrasing/818061/2 "2018-07-17T04:11:16Z")
I would read the first claim as: A is a necessary and sufficient condition for C, and also B is a necessary and sufficient condition for C. So if A is true, then C and B must be true as well. In the second claim, it’s possible that A is true but C and B are false. ---
**Author:** ![watchwolf49](https://avatars.discourse-cdn.com/v4/letter/w/e9c0ed/32.png) [@watchwolf49](https://boards.straightdope.com/u/watchwolf49)\ **Post date:** [July 17, 2018, 4:30am UTC](https://boards.straightdope.com/t/logic-math-phrasing/818061/3 "2018-07-17T04:30:25Z")
The first is a simple AND statement isn’t it? … both A and B must be true for C to be true, if either A or B are false, or both are false, then C is false … A **and** B are necessary … The second just seems to be adding words … it’s still A **and** B are necessary with the words “individually” and “jointly” being redundant … I would say both statements are logically equivalent … ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [July 17, 2018, 4:45am UTC](https://boards.straightdope.com/t/logic-math-phrasing/818061/4 "2018-07-17T04:45:53Z")
Symbolic notation is more transparent here; your sentence is equivalent to A ∧ B ↔ C ---
**Author:** ![Dr.Strangelove](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dr.strangelove/32/6613_2.png) [@Dr.Strangelove](https://boards.straightdope.com/u/Dr.Strangelove)\ **Post date:** [July 17, 2018, 5:11am UTC](https://boards.straightdope.com/t/logic-math-phrasing/818061/5 "2018-07-17T05:11:13Z")
> [@borschevsky](#): > > I would read the first claim as: A is a necessary and sufficient condition for C, and also B is a necessary and sufficient condition for C. That can’t be true unless A and B are the same. If A is a sufficient condition for C, then B can’t be a necessary condition for C. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [July 17, 2018, 5:14am UTC](https://boards.straightdope.com/t/logic-math-phrasing/818061/6 "2018-07-17T05:14:01Z")
> [@Dr.Strangelove](#): > > That can’t be true unless A and B are the same. Well, logically equivalent to each other. I think there’s more than one possible interpretation (as given by the first two replies), so the statement ought to be reworded for clarity. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [July 17, 2018, 2:18pm UTC](https://boards.straightdope.com/t/logic-math-phrasing/818061/7 "2018-07-17T14:18:27Z")
I would interpret the two phrasings the same, because while the first statement can be interpreted as meaning that A and B are equivalent, that’d be a really roundabout way of saying that, and so probably not what was meant. ---
**Author:** ![borschevsky](https://avatars.discourse-cdn.com/v4/letter/b/97f17d/32.png) [@borschevsky](https://boards.straightdope.com/u/borschevsky)\ **Post date:** [July 17, 2018, 4:11pm UTC](https://boards.straightdope.com/t/logic-math-phrasing/818061/8 "2018-07-17T16:11:33Z")
> [@DPRK](#): > > Symbolic notation is more transparent here; your sentence is equivalent to > > A ∧ B ↔ C If that’s what is meant, then I would have written the first claim as **A and B is a necessary and sufficient condition for C** , maybe including parens or quotes like **( A and B )** for clarity. Using “are” as in **A and B are … conditions** makes me read the and as a normal English and rather than the logical ‎∧, so I parse it as listing two statements that each individually have the given properties. I do see how people can read it the other way though. ---
**Author:** ![Lance\_Turbo](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lance_turbo/32/6156_2.png) [@Lance\_Turbo](https://boards.straightdope.com/u/Lance_Turbo)\ **Post date:** [July 17, 2018, 5:06pm UTC](https://boards.straightdope.com/t/logic-math-phrasing/818061/9 "2018-07-17T17:06:11Z")
> [@borschevsky](#): > > If that’s what is meant, then I would have written the first claim as **A and B is a necessary and sufficient condition for C** , maybe including parens or quotes like **( A and B )** for clarity. Using “are” as in **A and B are … conditions** makes me read the and as a normal English and rather than the logical ‎∧, so I parse it as listing two statements that each individually have the given properties. I do see how people can read it the other way though. As **Dr. Strangelove** stated earlier, “That can’t be true unless A and B are the same. If A is a sufficient condition for C, then B can’t be a necessary condition for C.” ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [July 17, 2018, 5:30pm UTC](https://boards.straightdope.com/t/logic-math-phrasing/818061/10 "2018-07-17T17:30:12Z")
If the original statement is really so unclear as to leave most readers scratching their heads or misinterpreting it, then maybe it is really a case of bad writing on the author’s part. Hopefully what actually holds true can be figured out from the context. PS > [@](#): > > As **Dr. Strangelove** stated earlier, “That can’t be true unless A and B are the same. If A is a sufficient condition for C, then B can’t be a necessary condition for C.” it could conceivably be that “A and B are necessary and sufficient conditions for C” is a poor way of writing: “the following are equivalent: (1) A; (2) B; (3) C.” I would go back to the original statements and work out for myself what actually holds. ---
**Author:** ![borschevsky](https://avatars.discourse-cdn.com/v4/letter/b/97f17d/32.png) [@borschevsky](https://boards.straightdope.com/u/borschevsky)\ **Post date:** [July 17, 2018, 5:31pm UTC](https://boards.straightdope.com/t/logic-math-phrasing/818061/11 "2018-07-17T17:31:03Z")
Yes, I understand that. I was talking about the phrasing of the claim, and not evaluating whether it’s a useful or correct statement. In any case, it’s perfectly reasonable to have three statements A, B, and C, which are all logically equivalent. A and B are then each individually necessary and sufficient conditions for C. ---
**Author:** ![OldGuy](https://avatars.discourse-cdn.com/v4/letter/o/3bc359/32.png) [@OldGuy](https://boards.straightdope.com/u/OldGuy)\ **Post date:** [July 17, 2018, 6:16pm UTC](https://boards.straightdope.com/t/logic-math-phrasing/818061/12 "2018-07-17T18:16:39Z")
Oh it was clear from the context that what was being stated was (A ∧ B) \<=\> C The second phrasing seem to me to say precisely that in the form (A ∧ B) =\> C, C =\> B, C =\> A Someone was arguing that the first phasing meant something different and I couldn’t follow their verbal reasoning. Thanks all. ---
**Author:** ![Dr.Strangelove](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dr.strangelove/32/6613_2.png) [@Dr.Strangelove](https://boards.straightdope.com/u/Dr.Strangelove)\ **Post date:** [July 17, 2018, 10:15pm UTC](https://boards.straightdope.com/t/logic-math-phrasing/818061/13 "2018-07-17T22:15:41Z")
> [@DPRK](#): > > it could conceivably be that “A and B are necessary and sufficient conditions for C” is a poor way of writing: “the following are equivalent: (1) A; (2) B; (3) C.” I would go back to the original statements and work out for myself what actually holds. It’s conceivable, but seems unlikely (I agree with **Chronos** here). It becomes even less likely as soon as you add a bit of concreteness and wiggle room for implicit conditions. For example, “chocolate sauce and whipped cream are necessary and sufficient toppings for an ice cream sundae”. It’s possible that I meant: (chocolate sauce is a necessary and sufficient topping for an ice cream sundae) AND (whipped cream is a necessary and sufficient topping for an ice cream sundae) Or expanded out even more: (chocolate sauce is a necessary topping for an ice cream sundae) AND (chocolate sauce is a sufficient topping for an ice cream sundae) AND (whipped cream is a necessary topping for an ice cream sundae) AND (whipped cream is a sufficient topping for an ice cream sundae) But that’s just obviously false, because some conditions are mutually exclusive. The much more reasonable interpretation is: (chocolate sauce AND whipped cream) are necessary and sufficient toppings for an ice cream sundae That is at least conceivably true. Note that the full conditions for C are inferred and not specified exactly, so we never make a direct logical equivalence between A\<=\>C or B\<=\>C. And it’s obvious that A and B are not logically equivalent. ---
**Author:** ![Hari\_Seldon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hari_seldon/32/5173_2.png) [@Hari\_Seldon](https://boards.straightdope.com/u/Hari_Seldon)\ **Post date:** [July 18, 2018, 1:04am UTC](https://boards.straightdope.com/t/logic-math-phrasing/818061/14 "2018-07-18T01:04:05Z")
I understood the first to mean that A and B are jointly necessary and sufficient for C and yes the two statements are equivalent. Perhaps the sentence could be interpreted as ambiguous, but the meaning seemed clear enough to me. If you interpret the sentence to mean A is equivalent to C and B is equivalent to C, then all three are equivalent and it is very unlikely you would put it that way, which is one reason that my original interpretation is the default.